关于第三阶WENO和CWENO重建在非均匀网格上的精度
On the accuracy of WENO and CWENO reconstructions of third order on nonuniform meshes
AI总结：
本文研究了非均匀网格上第三阶WENO和CWENO重建中参数ε的影响，探讨了ε作为局部网格大小函数的优化对误差和收敛阶的提升作用，并通过数值实验验证了不同ε选择在不同数值方案中的有效性。
摘要：
第三阶WENO和CWENO重建是数值方案中用于超几何守恒和平衡定律的高阶重建技术。在其定义中出现了一个小的正参数，通常称为ε，最初是为了避免在常数状态下的除以零问题，但其值后来被证明会影响方案的收敛性质。最近，两项关于均匀网格上ε作用的详细研究已发表。本文将这些结果扩展到非均匀网格上的有限体积方案，这对h自适应方案非常重要，展示了将ε作为局部网格大小h_j的函数的好处。特别是，我们展示了选择ε=h_j²或ε=h_j对误差和收敛阶是有益的，研究了在几种非均匀网格上这种选择对重建误差、线性输运方程的完全离散方案以及数值方案稳定性的影响。最后，我们比较了不同ε选择在圣维南系统浅水方程的平衡方案以及非线性守恒律方程的h自适应方案中的效果，并对CWENO重建的二维推广进行了数值测试。
英文摘要：
Third order WENO and CWENO reconstruction are widespread high order reconstruction techniques for numerical schemes for hyperbolic conservation and balance laws. In their definition, there appears a small positive parameter, usually called $ε$, that was originally introduced in order to avoid a division by zero on constant states, but whose value was later shown to affect the convergence properties of the schemes. Recently, two detailed studies of the role of this parameter, in the case of uniform meshes, were published. In this paper we extend their results to the case of finite volume schemes on non-uniform meshes, which is very important for h-adaptive schemes, showing the benefits of choosing $ε$ as a function of the local mesh size $h_j$. In particular we show that choosing $ε=h_j^2$ or $ε=h_j$ is beneficial for the error and convergence order, studying on several non-uniform grids the effect of this choice on the reconstruction error, on fully discrete schemes for the linear transport equation, on the stability of the numerical schemes. Finally we compare the different choices for $ε$ in the case of a well-balanced scheme for the Saint-Venant system for shallow water flows and in the case of an h-adaptive scheme for nonlinear systems of conservation laws and show numerical tests for a two-dimensional generalisation of the CWENO reconstruction on locally adapted meshes.
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